Answer
$\{(-3,0,2)\}$.
Work Step by Step
The given system of equations are.
$\Rightarrow \frac{x+3}{2}-\frac{y-1}{2}+\frac{z+2}{4}=\frac{3}{2}$
Multiply both sides by $4$.
$\Rightarrow 4\cdot \left (\frac{x+3}{2}-\frac{y-1}{2}+\frac{z+2}{4}\right )=4\cdot \frac{3}{2}$
Apply distributive property.
$\Rightarrow 2x+6-2y+2+z+2= 6$
$\Rightarrow 2x-2y+z+10= 6$
Add $-10$ to both sides.
$\Rightarrow 2x-2y+z+10-10= 6-10$
Simplify.
$\Rightarrow 2x-2y+z= -4$...... (1)
$\Rightarrow \frac{x-5}{2}+\frac{y+1}{3}-\frac{z}{4}=-\frac{25}{6}$
Multiply both sides by $12$.
$\Rightarrow 12\cdot \left (\frac{x-5}{2}+\frac{y+1}{3}-\frac{z}{4}\right )=-12\cdot \frac{25}{6}$
Apply distributive property.
$\Rightarrow 6x-30+4y+4-3z= -50$
$\Rightarrow 6x+4y-3z-26= -50$
Add $26$ to the both sides.
$\Rightarrow 6x+4y-3z-26+26= -50+26$
Simplify.
$\Rightarrow 6x+4y-3z= -24 $...... (2)
$\Rightarrow \frac{x-3}{4}-\frac{y+1}{2}+\frac{z-3}{2}=-\frac{5}{2}$
Multiply both sides by $4$.
$\Rightarrow 4\cdot \left (\frac{x-3}{4}-\frac{y+1}{2}+\frac{z-3}{2}\right )=-4\cdot \frac{5}{2}$
Apply distributive property.
$\Rightarrow x-3-2y-2+2z-6= -10$
$\Rightarrow x-2y+2z-11= -10$
Add $11$ to both sides
$\Rightarrow x-2y+2z-11+11= -10+11$
Simplify.
$\Rightarrow x-2y+2z= 1$...... (3)
Multiply equation (1) by $3$ and add to equation (2).
$\Rightarrow 3(2x-2y+z)+6x+4y-3z= 3(-4)-24$
Apply distributive property.
$\Rightarrow 6x-6y+3z+6x+4y-3z= -12-24$
Add like terms.
$\Rightarrow 12x-2y= -36$ ...... (4)
Multiply equation (1) by $-2$ and add to equation (3).
$\Rightarrow -2(2x-2y+z)+x-2y+2z= -2(-4)+1$
Apply distributive property.
$\Rightarrow -4x+4y-2z+x-2y+2z=8+ 1$
Add like terms.
$\Rightarrow -3x+2y= 9$ ...... (5)
Add equation (4) and (5).
$\Rightarrow 12x-2y-3x+2y= -36+9$
Simplify.
$\Rightarrow 9x= -27$
Divide both sides by $9$.
$\Rightarrow \frac{9x}{9}= \frac{-27}{9}$
$\Rightarrow x=-3$
Plug the value of $x$ into equation $5$.
$\Rightarrow -3(-3)+2y= 9$
Simplify.
$\Rightarrow 9+2y= 9$
Subtract $9$ from both sides.
$\Rightarrow 9+2y-9= 9-9$
Simplify.
$\Rightarrow 2y= 0$
Divide both sides by $2$.
$\Rightarrow \frac{2y}{2}= \frac{0}{2}$
Simplify.
$\Rightarrow y= 0$
Plug the values of $x$ and $y$ into equation (1).
$\Rightarrow 2(-3)-2(0)+z= -4$
Simplify.
$\Rightarrow -6+z= -4$
Add $6$ to both sides.
$\Rightarrow -6+z+6= -4+6$
Simplify.
$\Rightarrow z= 2$
The solution set is $\{(x,y,z)\}=\{(-3,0,2)\}$.
